Homology-changing percolation transitions on finite graphs

TL;DR

This paper studies how the homological properties of finite graphs change under percolation, identifying thresholds related to the emergence of non-trivial cycles and their dependence on graph growth and structure.

Contribution

It introduces a framework linking homological percolation thresholds to the growth of minimal non-trivial cycles on finite graph sequences.

Findings

Homology-changing thresholds depend on the growth of the smallest non-trivial cycle.

The giant cycle erasure threshold equals the percolation threshold under certain growth conditions.

In non-amenable cases, the thresholds differ, indicating a phase transition in homological properties.

Abstract

We consider homological edge percolation on a sequence $(\mathcal{G}_t)_t$ of finite graphs covered by an infinite (quasi)transitive graph $\mathcal{H}$, and weakly convergent to $\mathcal{H}$. Namely, we use the covering maps to classify $1$-cycles on graphs $\mathcal{G}_t$ as homologically trivial or non-trivial, and define several thresholds associated with the rank of thus defined first homology group on the open subgraphs. We identify the growth of the homological distance $d_t$, the smallest size of a non-trivial cycle on $\mathcal{G}_t$, as the main factor determining the location of homology-changing thresholds. In particular, we show that the giant cycle erasure threshold $p_E^0$ (related to the conventional erasure threshold for the corresponding sequence of generalized toric codes) coincides with the edge percolation threshold $p_{\rm c}(\mathcal{H})$ if the ratio $d_t/\ln n_t$ diverges, where $n_t$ is the number of edges of $\mathcal{G}_t$, and we give evidence that $p_E^0<p_{\rm c}(\mathcal{H})$ in several cases where this ratio remains bounded, which is necessarily the case if $\mathcal{H}$ is non-amenable.